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## 9. Miscellaneous Problems

1. #### Problem 9.1.

[P. Gauthier] Let be $K\subset\mathbb{C}$ be compact, let $\mathbb{C}\setminus K$ be connected, and let $f$ be a continuous function on $K$ which is holomorphic and non-zero on the interior of $K$. Is there a sequence of polynomials $\{p_n\}_{n=1}^\infty$ with no zeros in $K$ such that $p_n\to f$?
• #### Conjecture 9.2.

[J. Andersson, D. Farmer] Let be $K\subset\mathbb{C}$ be compact, $\mathbb{C}\setminus K$ connected, and $f$ be a continuous function on $K$ which is holomorphic and non-zero on the interior of $K$. Suppose $K\subseteq\{z:\frac{1}{2}<{\rm Re} z < 1\}$, then $\overline{d}\left( \left\{t\in\mathbb{R}: ||\zeta(\cdot + it) - f(\cdot)||_{\infty, K} < \epsilon\right\}\right) > 0,$ where

$\overline{d}(E):=\limsup_{T\to\infty} \frac{m(E\cap[0,T\,])}{T}.$
1. Remark. [P. Gauthier] Andersson has shown that a positive answer to Problem nonzero_approx and a confirmation of Conjecture conj_universal are equivalent.
• #### Problem 9.3.

[R. Pemantle] Let $\mu$ be a probability measure on $\{0,1\}^n$. Consider $f(\vec{z}\,) = \int {\vec{z}}^{\;\vec{\alpha}} d\mu(\vec{\alpha}\,)$ and $g(\vec{\lambda}\,) = \int e^{\;\vec{\lambda}\cdot\vec{\alpha}} d\mu(\vec{\alpha}\,).$ What property of $g$ is equivalent to the stability of $f$?
• #### Problem 9.4.

[G. Knese] Let $p(z)\in\mathbb{C}[z]$, $\deg(p)=n$. Suppose $p(z)\neq 0$ for all $|z|\le1$. Define $p^*(z):=z^n\overline{p\left(\frac{1}{\bar{z}}\right)}$. Then it is known that

$p(A)p(A)^\dagger \ge p^*(A)(p^*(A))^\dagger \;\; \textit{for any contractive matrix } A.$

($A$ is contractive means $\sup\limits_x \frac{||Ax||_2}{||x||_2}<1.$) In 2D, we replace $A$ with a pair of commuting contractions and $p$ with a bivariate polynomial with no zeros in $\{z:|z|\le 1\}\times\{w:|w|\le 1\}$. The 3D generalization fails – when does it hold?
• #### Problem 9.5.

[P. Brändén] Find more examples and attempt to characterize Markov processes preserving stability. Explicitly, let $\psi:\mathbb{R}_{MA}[x,y,z]\to\mathbb{R}_{MA}[x,y,z]$ be a linear operator. When is $e^{t\psi}:\mathbb{R}[x,y,z]\to\mathbb{R}[x,y,z]$ stability preserving for all $t\ge0$ ?

Cite this as: AimPL: Stability and hyperbolicity, available at http://aimpl.org/hyperbolicpoly.